Some maximality results for effect-valued measures

We give an approach to the theory of effect-valued measures taking their values in the positive operators on a Hilbert space. The concept of operator-valued measure is fundamental in modern theories of quantum measurements. In the paper we introduce and study relations of dominance and equivalence between two effect-valued measures and concepts of maximal and minimal effect-valued measures. Characterizations of maximal effect-valued measures are obtained in the discrete case and in the case of commutative range. As an example we study the so-called Bargmann measure which can be interpreted as a simultaneous non-ideal measurement of position and momentum.